Analysis of Bi-directional Functionally Graded Porous Beams Vibrations Utilizing Ritz Solutions and Higher-Order Shear Deformation Theory

This paper presents a novel hybrid function for analyzing the free vibration of bi-directional functionally graded porous (2D-FGP) beams. The higher-order shear deformation theory (HSDT) serves as the foundation for the development of the displacement field of the 2D-FGP beams. The variation in material along the x- and z-axes within the beam volume is determined by a power-law relationship. The porosity of the functionally graded material is represented by three different distributions. By using Lagrange’s equation, the governing equations can be inferred. To define the displacement field, a hybrid function that combines an exponential-trigonometric function is devised. Different boundary conditions (simply-supported, clamped-clamped, and clamped-free), gradation exponents in the x and z directions, the porosity coefficient, the porosity distribution types (FGP-1, FGP-2, and FGP-3), and the slenderness ratio are utilized to calculate the frequency. A study of finite element method results shows that the suggested method is correct and works well.